Is there any way I can add a payment method even though I’m no the family organizer?

If the Family Sharing group organiser has turned on purchase sharing you will need to have them turn it off to make use of your own payment method. See How to share purchases with your family, and specifically Turn off purchase sharing which has these steps for a family organiser:

1. See your Purchase Sharing settings in Settings on your iPhone or iPad or in System Preferences on your Mac.
2. Turn off purchase sharing:

• On your iPhone, iPad, or iPod touch, tap Stop Purchase Sharing.
• On your Mac, click Turn Off, then click Stop Purchase Sharing.

Each member of the family can set up a credit card for other purchases that they make on their iOS device, such as purchases made on the web from vendors like Amazon, etc. but they can’t use the card for App Store, iTunes and Subscription using Family Sharing. All of those purchase must go through the Family Organizer who agrees to make the payments as the organizer of the plan.

There is an exception to this and it would be if each member redeems Apple Gift cards. The balance on their gift cards will used to pay for purchase before the Family organizer’s credit card. My wife, my daughter and I all redeem gift cards so that my debit card is almost never used for purchases. The only time that my card has been used in the 3-1/2 years of Family Sharing is when my daughter was pennies short on a couple of songs that she bought.